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%I M3863 N1584 #54 Feb 02 2022 04:52:12
%S 5,15,55,225,979,4425,20515,96825,462979,2235465,10874275,53201625,
%T 261453379,1289414505,6376750435,31605701625,156925970179,
%U 780248593545,3883804424995,19349527020825,96470431101379,481245667164585,2401809362313955,11991391850823225
%N a(n) = 1^n + 2^n + ... + 5^n.
%C a(n)*(-1)^n, n>=0, gives the z-sequence for the Sheffer triangle A049460 ((signed) 5-restricted Stirling1 numbers), which is the inverse triangle of A193685 (5-restricted Stirling2 numbers). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The a-sequence for each (signed) r-restricted Stirling1 Sheffer triangle is A027641/A027642 (Bernoulli numbers). - _Wolfdieter Lang_, Oct 10 2011
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001552/b001552.txt">Table of n, a(n) for n = 0..200</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=365">Encyclopedia of Combinatorial Structures 365</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%F a(n) = Sum_{k=1..5} k^n, n >= 0.
%F O.g.f.: (5 - 60*x + 255*x^2 - 450*x^3 + 274*x^4)/Product_{j=1..5} (1 - j*x). - _Simon Plouffe_ in his 1992 dissertation
%F E.g.f.: exp(x)*(1-exp(5*x))/(1-exp(x)) = Sum_{j=1..5} exp(j*x) (trivial). - _Wolfdieter Lang_, Oct 10 2011
%t Table[Total[Range[5]^n], {n, 0, 40}] (* _T. D. Noe_, Oct 10 2011 *)
%o (PARI) a(n)=if(n<0,0,sum(k=1,5,k^n))
%o (Sage) [3**n + sigma(4, n) + 5**n for n in range(22)] # _Zerinvary Lajos_, Jun 04 2009
%o (Sage) [1 + 2**n + 3**n + 4**n + 5**n for n in range(22)] # _Zerinvary Lajos_, Jun 04 2009
%Y Column 5 of array A103438.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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